Modular Functions and Holonomic Functions

Holonomic (also called: D-finite) functions satisfy linear differential equations with polynomial coefficients-in contrast to modular functions.

Holonomic functions and the coefficient sequences of their Taylor series are ubiquitous objects in enumerative combinatorics. The coefficients of the Fourier expansions of modular functions often carry interesting number theoretic information. For example, the sequence p(4)=5, p(9)=30, p(14)=135, etc. are partition numbers which, as observed first by Ramanujan, are all divisible by 5.

The talk investigates various connections between holonomic and modular functions. To this end, computer algebra is used.